If $H$ is a subgroup of a group $G\text{,}$ then right cosets are not always the same as left cosets; that is, it is not always the case that $gH = Hg$ for all $g \in G\text{.}$ The subgroups for which this property holds play a critical role in group theory—they allow for the construction of a new class of groups, called factor or quotient groups. Factor groups may be studied directly or by using homomorphisms, a generalization of isomorphisms. We will study homomorphisms in Chapter 10.